Proof of the entropic c-theorem
Abstract
We prove an entropic c-theorem in two dimensions by constructing a c-function from the mutual information between spacelike separated wedges, defined through the relative entropy. The proof follows Casini and Huerta's, with strong subadditivity of entanglement entropy replaced by increasing differences of the mutual information, which we prove directly in the continuum from the split property and standard properties of relative entropy. The benefit of relative entropy is that every quantity we use is finite and no cutoff ever needs to be introduced. The assumptions of the theorem are only the standard axioms of algebraic quantum field theory, the split property with finite mutual information across a gap (both implied by modular p-nuclearity), and a finite ultraviolet bound on the resulting c-functions. We also note that, formally, the c-function is proportional to the minimal averaged null energy of a state that reproduces the vacuum correlations between the two wedges.
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BibTeX
@misc{kudlerflam2026proof,
title = {{Proof of the entropic $c$-theorem}},
author = {Jonah Kudler-Flam},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40874},
eprint = {2610.40874},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40874},
}
RIS
TY - UNPB
TI - Proof of the entropic c-theorem
AU - Jonah Kudler-Flam
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40874
UR - https://arxiv.org/abs/2610.40874
AB - We prove an entropic c-theorem in two dimensions by constructing a c-function from the mutual information between spacelike separated wedges, defined through the relative entropy. The proof follows Casini and Huerta's, with strong subadditivity of entanglement entropy replaced by increasing differences of the mutual information, which we prove directly in the continuum from the split property and standard properties of relative entropy. The benefit of relative entropy is that every quantity we use is finite and no cutoff ever needs to be introduced. The assumptions of the theorem are only the standard axioms of algebraic quantum field theory, the split property with finite mutual information across a gap (both implied by modular p-nuclearity), and a finite ultraviolet bound on the resulting c-functions. We also note that, formally, the c-function is proportional to the minimal averaged null energy of a state that reproduces the vacuum correlations between the two wedges.
ER -